# Properties

 Label 2-140-7.4-c1-0-0 Degree $2$ Conductor $140$ Sign $0.701 - 0.712i$ Analytic cond. $1.11790$ Root an. cond. $1.05731$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (2.5 − 0.866i)7-s + (1 + 1.73i)9-s + (−3 + 5.19i)11-s + 2·13-s − 0.999·15-s + (3 − 5.19i)17-s + (−4 − 6.92i)19-s + (−0.500 + 2.59i)21-s + (−1.5 − 2.59i)23-s + (−0.499 + 0.866i)25-s − 5·27-s + 3·29-s + (−1 + 1.73i)31-s + ⋯
 L(s)  = 1 + (−0.288 + 0.499i)3-s + (0.223 + 0.387i)5-s + (0.944 − 0.327i)7-s + (0.333 + 0.577i)9-s + (−0.904 + 1.56i)11-s + 0.554·13-s − 0.258·15-s + (0.727 − 1.26i)17-s + (−0.917 − 1.58i)19-s + (−0.109 + 0.566i)21-s + (−0.312 − 0.541i)23-s + (−0.0999 + 0.173i)25-s − 0.962·27-s + 0.557·29-s + (−0.179 + 0.311i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$140$$    =    $$2^{2} \cdot 5 \cdot 7$$ Sign: $0.701 - 0.712i$ Analytic conductor: $$1.11790$$ Root analytic conductor: $$1.05731$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{140} (81, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 140,\ (\ :1/2),\ 0.701 - 0.712i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.02035 + 0.427612i$$ $$L(\frac12)$$ $$\approx$$ $$1.02035 + 0.427612i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + (-0.5 - 0.866i)T$$
7 $$1 + (-2.5 + 0.866i)T$$
good3 $$1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2}$$
11 $$1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 - 2T + 13T^{2}$$
17 $$1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 - 3T + 29T^{2}$$
31 $$1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + 3T + 41T^{2}$$
43 $$1 - 5T + 43T^{2}$$
47 $$1 + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 3T + 83T^{2}$$
89 $$1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + 10T + 97T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−13.36706048412841757629063522976, −12.22731696367484953935176051160, −10.90730112731691485270055581781, −10.49487441071752086039410885698, −9.338167679084910960978160382074, −7.82713327274107256128074261923, −6.99365084362842027402329299925, −5.17040697582510466059222722227, −4.48615001777838913573719204557, −2.30107921941669512543073014083, 1.50858264136031691399360890224, 3.72731121181989691548103376156, 5.53064951770157521542506418499, 6.20864127638802650703356683793, 8.064615327217175668888713984766, 8.444227188983535485232190633415, 10.10354980205228988436042381203, 11.08619625972759325409613725196, 12.12519234019566390850642585439, 12.92537522565612462755954713376